1
THAT’S ODD!
Odd
The word odd has taken anything but a straight path on its journey to the twentyfirst century.
The Old English word odde meant an angle, a point or the tip of a weapon from
whence it took on the meaning of the point of a triangle and then, figuratively, a
third man having the casting vote in a dispute. The word umpire is derived from
non-pair (not even) or the odd man who takes on the task of being a referee.
By extension from its connection with the number three, odde came to connote
any uneven number, namely, a whole number which when divided by two leaves
a remainder. Reduced to a simple diagram this could be represented thus:
Fig. 1
– where the remainder “sticks out”. This sense of the word odd is recorded as
early as 1375.
The word in its application to individuals came to describe someone who is
unusual or “sticks out” from the rest or behaves strangely. This is the sense in
which Shakespeare uses it in Love’s Labour’s Lost:
“He is too picked, too spruce, too effected, too odd …” (Act 5 Scene 1)
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However, neither of the phrases odds bodykins or od’s bodikin in Hamlet (Act
2 Scene 2) nor similar oaths originate from the word odde. They derive from
foreshortening the phrase “God’s body” in order to avoid profanity. Another
example of this is the word zounds or “God’s wounds” (Othello Act 1 Scene 1).
The term odds, indicating a difference, first appeared in 1542. Shakespeare used
the word to indicate variously: superiority, advantage, inequality, differences,
quarrels and probability.
Odds is now used mainly as a betting term to indicate the proportion in which
the amount staked bears to the amount which might be won and we speak of
“taking odds” and “laying odds”. We also use phrases such as “it is odds on that”,
or “the odds are that”, both of which refer to the chance or probability that a
specific event will happen. We also use the phrase “over the odds” meaning in
excess of the going rate.
In addition to its significance as a mathematical term, we use odd in many
different phrases such as:
• doing things at odd moments or unplanned intervals;
• the odd trick, being the last and deciding trick at Whist;
• odd coins, indicating a small amount of cash left over – odd being slang in
Ireland for loose change;
• at odds, meaning at variance with somebody or something;
• odds and ends and its extension during World War II to odds and sods which
Eric Partridge in A Dictionary of Forces Slang 1939-1945 (Secker & Warburg)
defines as “men on miscellaneous duties”;
• odd jobs, unconnected tasks done on a casual basis;
• an odd lot, a random mixture of goods;
• an odd thing, something rare or unusual;
• an oddball, US slang for an eccentric person or outsider;
• odd as in “odd socks”. (The mystery surrounding the regular disappearance
of one of a pair of socks after a machine-wash has, incidentally, been solved
by a reader of The Times who, many years ago, replied to a letter bemoaning
such a loss from another reader: “I know where his odd sock is. It is in my
washing machine.”)
What most of these usages have in common is an allusion to “sticking out”,
harking back to the very origin of the word odd so that, after all of its journeying, the word has kept its faith.
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ODD WORDS, EVEN NUMBERS
How odd it is that the word odds, though formed by pluralising odd, is itself
a singular noun. Other such nouns with a similar treatment are trousers, jeans
and news as to which see page 122.
The conjunction of the words odd and even throws up some surprises. Harking
again back to Shakespearian times, odd-even (Othello Act 1 Scene 1) meant
around midnight whereas in nuclear physics the phrase now refers to nuclei
of odd mass number and of even mass number. The apparently oxymoronic
phrase even odds (often referred to as even money) has an entirely different
connotation referring to a bet which, if successful, pays winnings equal to the
amount staked as well as the return of that stake.
Pythagoras wrote about the concept of odd and even numbers in Greece as long
ago as 500BC and Euclid defined an even number around 300BC as “a number
divisible into two parts”.
A more up-to-date definition of an odd number is an integer (see page 24) that
cannot be divided by 2 without a remainder. In mathematical terms, an odd
number is a number having the pattern of 2n+1 where n is an integer. This
makes it easy to understand why zero is treated as an even number.
More even than odd
Why do we treat even in all its manifestations as more “favourable” than odd?
It is possible that the inherent nature of mathematical calculations favours
evenness in that the result of multiplying an even number by an odd number (or
by any number of odd numbers) results in an even number.
Also, of the three alternative ways of adding together odd and even numbers (O
+ E, O + O and E + E), two result in an even result and only one in an odd result.
However, there are four alternatives since E + O should be included. Indeed,
if we include subtraction and thus count all eight combinations of adding and
subtracting two numbers, we see that there is no imbalance since there are as
many odd results as there are even:
O + E = Odd
E + O = Odd
O + O = Even
E + E = Even
O - E = Odd
E - O = Odd
O - O = Even
E - E = Even
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A determining factor in the preferential status of even numbers arises from the
fact that our system is a decimal system based on 10 and that each threshold
reached in the process of counting (10, 20 to 90, 100, 1,000 and so on) is an even
number.
However, evenness has not always been in the ascendancy. Plutarch (ca. AD
46-120) regarded the number 1 as having divine attributes. Virgil (70-19BC)
referred to odd numbers being “to the gods delightful” and in Shakespeare’s
Merry Wives of Windsor Falstaff says to Mistress Quickly: “I hope good luck
lies in odd numbers... They say there is divinity in odd numbers.” The number 1
continues to give rise to curiosity. Benford’s Law, conceived by Frank Benford
(1883-1948), tells us that in tables of naturally occurring numbers such as the
highest mountains and the populations of the counties of England, the number
1 occurs as the first or leading digit with a frequency often greater than 30 per
cent. However, statistically, we would expect each of the digits 1 to 9 to occur
with a frequency of only 11.1 per cent, that is, 100⁄9. A leading 9 is the least likely
to occur. This is a phenomenon which has been hard to explain but is used to
detect fraud since, where figures inserted in a bogus set of accounts fail by a
significant margin to match the expectations of Benford’s Law, there must be a
suggestion that those accounts are fictional.
In contrast, the number 2 is seen as having characteristics opposed to divinity.
Indeed, deuce is still used as a euphemism for the devil in phrases such as “what
the deuce?” and “a deuce of a mess”. How has deuce come to mean the devil?
In a game using two dice, the word deuce has come to indicate the lowest
and unluckiest score – possibly eliciting an exclamation of annoyance from its
thrower. It is not difficult to see how such expression of annoyance (possibly
influenced by duus, Low German for “the devil”) came to have an extended
meaning – where the devil might be regarded as to blame for that unlucky throw.
It is not clear when the number 13 (see page 108) first became associated with
bad luck but it is possible that the seeds of this superstition can be found in the
Last Supper and its 13 participants. Since then, the disenchantment with that
particular odd number has led to our current fashionable preference for even
numbers, encouraged no doubt by the positive notions conveyed by qualities
such as even-handedness and even-temperedness.
We talk of impairment meaning “damage” or “loss” where pair means “even”,
thus giving rise to yet further conditioning.
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ODD WORDS, EVEN NUMBERS
Boustrophedon
Separating odd numbers from even numbers occurs on a daily basis with street
numbering. We are familiar with the usual method of numbering houses and
shops where they bear consecutive odd numbers on one side of the street and
consecutive even numbers on the other side. However, that is not universal
practice.
The numbering of some streets starts on one side using the consecutive numbers
1, 2, 3, 4 etc. and, at the end of the road, continues crossing over and doubles
back on itself with a continuation of that series. We can refer to such a street
as boustrophedon, meaning “as the ox ploughs” in reference to the fact that,
on such a street, the numbering crosses back on itself resembling the pattern
followed by an ox when ploughing a field.
Squaring the odds
Consecutive odd numbers starting with 1 have the remarkable quality that,
wherever we stop counting, their aggregate at that point will always result in
a square number (a number multiplied by itself once). Thus, the addition of the
first two odd numbers, 1 and 3, comes to 4 which is 2 x 2 or 22 and 1 + 3 + 5 + 7 +
9 equals 25 or 52 and so on. Why does this happen? The phenomenon can most
easily be explained by an extension of the diagram in Fig. 1 on page 15 using a
pyramid of boxes where every successive tier involves adding two extra boxes
as follows:
1
3
5
7
9
Total = 25
Fig. 2
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If we move the lighter shaded boxes in Fig. 2 and transpose them to the other
side of the figure, we form a complete square as follows:
1
2
3
4
5
1
2
3
4
5
Total
5 x 5 = 25
Fig. 3
Cubes
We find a slightly different pattern involving consecutive odd numbers when
we look at cubes (cube – a number multiplied by itself twice):
13 = 1 = 1
23 = 8 = 3 + 5
33 = 27 = 7 + 9 + 11
43 = 64 = 13 + 15 + 17 + 19 and so on.
A similar pattern to the square pattern occurs with numbers to the power of 4:
14 = 1 = 1
24 = 16 = 1 + 3 + 5 + 7
34 = 81 = 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 and so on.
Left-hand side page, right-hand side page
A maths teacher was writing a book dealing with various mathematical concepts,
one of which was the difference between odd numbers and even numbers.
On one page he planned to explain the concept of an odd number and, on the
facing page, the concept of evenness.
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ODD WORDS, EVEN NUMBERS
The teacher was eager to ensure that the left-hand side page was numbered
as an odd page in keeping with its subject matter and that the right-hand side
should, correspondingly, have an even number as its page number. The odd
entry had to precede the even entry. However, the teacher had to abandon this
plan. Why? In any printed book or pamphlet with two-sided text, the page on
the left-hand side is referred to as the verso (from Latin vertere, to turn) and
the right-hand side page is the recto (from Latin rectus, right). The convention
(or, more accurately, the invariable practice) is to number the verso page as an
even number and the recto page as an odd number. This arises as a result of
the printer’s practice of starting each chapter of a book on a recto page so that
it will bear a page number which is odd. This practice of starting on a recto
page may have grown not merely as a result of convention but to facilitate the
production by printers of individual chapters or pamphlets since if each chapter
of any printed matter begins on a recto there would be no need for the printer
to renumber its pages.
If we envisage an offprint, it becomes clear that page 1, the recto, is the front
page or the “face” of the document and page 2, over the page, is the “back” side
of the front page.
If this hallowed practice were to be followed (as the teacher’s publisher insisted),
the verso would have had to have been an even number page and the recto an
odd number with the result that it would not have been possible to meet the
teacher’s requirements.
To ponder…
Question A
At precisely 19.59 on 19 November 1999 there occurred an event which will not
occur again for more than a thousand years. To be precise, this event can next
be expected to occur at 1.11 on 1 January 3111. What is that event?
Clue 1: The self-same event in 1999 occurred in every time zone.
Clue 2: Try setting out the dates in numbers only.
The answer is on page 209.
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Question B
A parallel event occurred at 22.48 on 28 August 888 which did not again occur
until 00.00 on 2 February 2000. What is that event?
The answer is on page 209.
Question C
Odd will always be odd in that it bears an odd number of letters while even
contains an even number of letters; but what is “never odd or even”?
Hint: Focus on the words of the question rather than on anything mathematical.
See page 209 for the answer.
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ODD WORDS, EVEN NUMBERS
2
NUMBERS, NUMERALS,
INTEGERS, DIGITS, FIGURES,
UNITS and AND
Some distinctions
In considering basic mathematical processes we refer to a variety of different
but overlapping terms. We refer variously to numbers, numerals, integers, digits,
figures and units. What does each of them mean?
Let us start with the question as to whether there is any difference in meaning
between number and numeral – two words which in common parlance are often
used as alternatives. We speak, for example, both of “Roman numbers” and
“Roman numerals”.
An understanding of the difference between the two words is perhaps best facilitated by looking at number as an abstract concept; that is to say, as having no
material existence. Number imparts the idea of counting things or of quantity.
The word might also signify the order in which things occur. In each case, we
are juggling with a concept rather than with something concrete.
In contrast, the word numeral indicates a symbol which represents or denotes
a particular number. Such a symbol is not abstract. It may consist of the Roman
numeral M, representing 1,000, the Arabic 7 or any of the host of symbols used
in other languages and systems to represent numbers. A numeral may have
existence on the printed page or a computer or some digital read-out or it may
present itself as a house number to be attached to a front door. A numeral is
thus any mark or figure or symbol which represents a number or is used to
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express a number – particularly a single digit. In each such case, it is capable of
being perceived.
This is not to say that the words number and numeral cannot, in many circumstances, be used interchangeably. Both words come from the Latin numerus,
meaning “a number” or “reckoning”, and from the Indo-European root nem,
meaning “to apportion, divide, take or allot”. Hence, we have words such as
nomad (allotted land as pasture), nome (a province of Greece), nomarch (a
senior Greek administrator whose title is easily confused with monarch) and a
host of words with the -nomy suffix such as agronomy, astronomy, economy,
gastronomy and taxonomy. We also get nemesis (a distribution of what is due).
An integer (pronounced with the emphasis on the first syllable and with a soft
g) is any whole number including a negative number (and also zero). Integer
has an interesting Indo-European root tag, to touch, as in the children’s game
of the same name. Combined with the Latin prefix, in-, meaning “not”, this root
provides us with the concept of being intact or entire. Hence, integer refers to
an entire or a whole number. From tag we also derive words such as tangent,
tactile, tact, taste, tangible, contact, contagious and tax (touches everyone).
The word digit from the Latin digitus, meaning “finger”, is used to signify the
first nine whole numbers and zero, resonating with man using his fingers to
count. Hence, the phrase “single digits” is strictly tautologous. Digit derives from
the Indo-European root deik, meaning “to show or utter”, from which we get
words such as indicate, edict, judicial, prejudice, dictate, abdicate and vindicate.
Figure comes from the Latin figura, meaning “a simple shape or form”. Figure
has many meanings. In mathematics it indicates a whole number and is often
interchangeable with the word digit. It is also often used interchangeably with
number, particularly in colloquial expressions such as “I am no good at figures”
and “figuring something out”. Figure, too, has an interesting Indo-European
ancestor – the root dheigh, meaning “to form, knead or shape”. Hence, we get
dig, ditch, dough and lady or la-dy being a loaf-kneader (OE Llaef-diger) and
perhaps also paradise, meaning “an enclosed or shaped park or garden”.
Unit from the Indo-European root oino (see one on page 133) indicates, in
mathematics, a single number as in “tens and units” – that is, in any number, the
figure found on the extreme right.
Another instance where we have a choice of word occurs with the different
expressions we regularly use to signify the process of adding one quantity to
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ODD WORDS, EVEN NUMBERS
another. We variously employ several terms for this function – we talk of “9 add
6”, “9 plus 6”, “9 more than 6” and “9 and 6”. Where do these terms come from
and do they have any differences in meaning?
Add comes from the IE do, meaning “to give”, from which we get words such
as donation and dosage and the L do, dare, datum from which, in turn, we get
the L addere, meaning “to join or unite”, and its gerundive, addendum, meaning
“that which should be added”.
Plus originates from the IE ple (or pel), meaning “full, much or many”. From
ple we derive words such as plenty, plethora, plenary, plebeian, plebiscite, plural
and replenish, as well as surplus, the prefix poly- and the conjunction plus from
the L meaning “more”. When we add to something the effect is to inflate it or
make it fuller.
However, the precise meanings which we attribute to this everyday word (and
its sister, L minus, meaning “less”) in a mathematical context are not clear.
Dictionary definitions of plus abound including “combined with”, “in addition
to”, “added to”, “increased by”, “to which is added”, “more”, “more than” and
a “summation of”. Although we might expect to be influenced by the L plus
meaning “more”, this sense creates a syntactical difficulty since, in the phrase
“9 plus 7 equals 16”, the substitution of more for plus does not make sense. To
incorporate the concept of more we would have to change the phrase to say that
“16 is more than 9 by 7”.
The fact that in the phrase “9 minus 7” we can indeed, without loss of mathematical or grammatical sense, substitute less for minus does not solve the dilemma
as regards plus, particularly when there are several perfectly consistent formulations which meet the requirements of both syntax and consistency between
plus and minus.
Treating plus, in a mathematical context, as meaning “in addition to” and minus
as meaning “with the subtraction of” fits the bill. Alternatively, we might translate these terms respectively as meaning “increased by” and “reduced by” in
each case without any ambiguity.
This brings us to more, from the IE me, great, from which we derive words
such as magnify, magnate, master, magistrate, magnificent and majesty. We also
derive the prefix mega – from the same root. The phrase “9 more than 7” is thus
merely a synonym for “9 greater than 7”.
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And possibly comes from the IE ant, meaning “against, opposed or in front
of”, providing us with words such as antagonism, antipathy and antipodes. This
short word thus has a more complex meaning than we might expect from a
mere conjunction. When used, for example, in the title of Bertrand Russell’s
Freedom and Organisation it has the function not merely of linking the two
concepts but of indicating that they are opposed to each other. The word and, of
course, serves mainly as a conjunction between words and phrases. However,
its pithiness makes it useful not only as a language tool but also in simple mathematics where, having the addition process in mind, we might colloquially
enquire “What is 7 and 9?”
Curiously, there is one particular respect in which we treat and, when used in
a mathematical sense, differently from the way in which we use it as a simple
conjunction in everyday speech. When used as a conjunction between two nouns
in ordinary speech, and commands a plural verb so that we say “Groucho and
Harpo were brothers” and “time and tide wait for no man”. There is nothing
strange in that. However, when we give expression to a phrase using and in a
mathematical context, we use the singular so that we say “7 and 9 equals 16”
and “7 and 9 comes to 16”. This is because we are not treating the 7 and the 9 in
this context as two separate items but are focusing on the phrase “7 and 9” as
a discrete and singular constituent of the first part of a mathematical equation.
Calculation
The words considered in this chapter have the common factor that they each
connote or anticipate some form of arithmetic calculation – another word with
an unexpected etymology.
The word calculation comes from the Latin calx, meaning “chalk” or “limestone”.
In ancient Rome the facility for working out arithmetic calculations, such as
we would now carry out with the use of pencil and paper, did not exist – nor
could the manipulation of Roman numerals ever have been a straightforward
exercise. However, the ancient Romans coped by using a counting board (a form
of abacus) containing grooves on which small chalk pebbles could be placed
and moved around for the purposes of indicating numbers and processing
arithmetic operations.
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ODD WORDS, EVEN NUMBERS
The Latin for such a pebble was calculus, the diminutive form of calx, and the
origin both of a branch of mathematics with the self-same name as well as the
word calculation and its cognates such as calculate, calculator and incalculable.
It is no wonder that the image of the scientist busily chalking up equations on a
blackboard is such a powerful one.
Counting
There are many words which commence with or include count.
The verb to count, meaning “to calculate or reckon”, has given us counter, being
a disc or object used in counting, and counter meaning “the surface on which the
process of counting takes place”. We also have words such as account, discount,
and compute. Each of these derive from the L computare, to calculate, and the OF
conte which, in the sixteenth century, came to be spelt compte, after the Latin,
and so gave rise to the word comptroller, an erroneous spelling of controller,
which nonetheless remains in use today in titles such as the Queen’s Comptroller
of the Household, an ancient position in the British royal household.
The word counter, meaning “opposite or opposed”, comes from the OF contre
and as a prefix generates scores of words including counterfeit, counterfoil,
countermand and countersign – none of which has any etymological connection
with the mathematical count although there are some respects in which a reckoning of some sort might be involved.
Count, indicating a member of the nobility, originated from yet another source.
It comes from the French comte (with an m rather than an n) which, in turn,
derives from the L comitare, meaning ”to accompany”, referring to a companion
or part of the retinue of a member of royalty or of some other distinguished
person. The land acquired by a count thus became known as his county.
However, counterintuitively, county is not cognate with country which probably
has its origins in the L contrata regio (from contra, meaning “opposite”) indicating the region spread out before one.
Countenance, meaning “demeanour, aspect or appearance”, has yet another
origin. It derives from the French contenir, to maintain (oneself).
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